Answer to Question #190986 in Calculus for Abhishek Gavhane

Question

The coefficient of x in the Maclaurin's expansion of sinx sink is

Answer 1

Given, $f(x)=sinx sink$

so

$f'(x)=cosx sink\Rightarrow f'(0)=cos0 sink=sink \\ f''(x)=-sin xsink\Rightarrow f''(0)=0 \\ f'''(x)=-cosxsink\Rightarrow f'''(0)=-sink$

Maclaurins expansion is given by-

$f(x)=f(0)+xf'(0)+\dfrac{x^2}{2!}f''(0)+\dfrac{x^3}{3!} f'''(0)... \\[9pt] f(x)=0+x(sink)+\dfrac{x^2}{2!}\times 0+\dfrac{x^3}{3!} (-sink)$

The coefficient of x in the expansion is sink.